Factoring Concepts and Methods

Overview

This lecture introduced the concept of factoring, focusing on identifying and using the Greatest Common Factor (GCF) in polynomials, including numbers and variables, and demonstrated factoring by grouping for complex expressions.

Understanding Factors

• A factor is a number or variable that divides another exactly (without remainder).
• Factors multiply to create terms; only multiplication, not addition, counts in factoring.
• Example: Factors of 15 include 3 and 5 because 3 × 5 = 15.
• Variables can also be factored, e.g., x³ × x⁴ = x⁷.

Common and Greatest Common Factors (GCF)

• Common factors are factors shared by two or more terms.
• The GCF is the largest factor (number or variable) common to all terms.
• To find the GCF of numbers, list all factors and find the largest common one.
• To find the GCF with variables, use the smallest exponent present in all terms.

Factoring Numbers and Variables Together

• GCF for terms with numbers and variables: determine the GCF for numbers, then for each variable, and combine.
• E.g., GCF of 9x⁴, 15x, 6x² is 3x.

Methods to Find GCF

• Method 1: List all factors and compare.
• Method 2: Prime factorization—break each term into primes and take the lowest powers shared.
• Relatively prime numbers share only 1 as a factor.

Factoring Polynomials by the GCF

• Always begin factoring by finding and extracting the GCF from all terms.
• Steps:
  1. Find the GCF of all terms.
  2. Write the GCF outside parentheses.
  3. Divide each term by the GCF and write results inside parentheses.
  4. Check work by distributing (should match the original).

Factoring Expressions with Fractions

• When all terms are fractions with the same denominator, factor out the denominator as part of the GCF.

Factoring by Grouping

• Used when expressions have four terms and no GCF for all terms.
• Group terms so each pair has a common factor, factor out those GCFs, and look for a common binomial to factor further.
• If rearrangement is needed for successful grouping, ensure paired terms share a factor.

Special Considerations in Factoring

• Always factor out a negative if it results in the leading term of the parentheses being positive.
• After factoring, ensure terms inside parentheses are relatively prime.

Key Terms & Definitions

• Factor — A number or variable that divides another without a remainder.
• Common Factor — Shared factor of two or more numbers or variables.
• Greatest Common Factor (GCF) — The largest shared factor among terms.
• Relatively Prime — Two terms with only 1 as a common factor.
• Prime Factorization — Expressing a number as a product of prime numbers.
• Factoring by Grouping — A method for factoring four-term polynomials by grouping pairs with common factors.

Action Items / Next Steps

• Practice GCF factoring with provided example problems.
• Review steps for factoring by grouping.
• Prepare for the next lecture on factoring trinomials (expressions with three terms).